Stable Homotopy Groups of Spheres: a Computer-Assisted Approach 4

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MATHEMATICS Table 1. Stable homotopy groups in dimensions 1 to 61

k v1-torsion at the prime 2 v1-torsion at odd primes v1-periodic Group of smooth structures 1 ·· 2 · 2 ·· 2 · 3 ·· 8·3 · 4 · · · ? 5 · · · · 6 2 · · ·

7 · · 16·3·5 b2 8 2 · 2 2 9 2 · 22 2·22 10 · 3 2 2·3

11 ·· 8·9·7 b3 12 ···· 13 · 3 · 3 14 2·2 · · 2

15 2 · 32·3·5 b4·2 16 2 · 2 2 17 22 · 22 2·23 18 8 · 2 2·8

19 2 · 8·3·11 b5·2 20 8 3 · 8·3 21 22 · · 2·22 22 22 ·· 22

23 2·8 3 16·9·5·7·13 b6·2·8·3 24 2 · 2 2 25 ·· 22 2·2 26 2 3 2 22·3

27 · · 8·3 b7 28 2 ·· 2 29 · 3 · 3 30 2 3 · 3 2 2 31 2 · 64·3·5·17 b8·2 32 23 · 2 23 33 23 · 22 2·24 34 22·4 · 2 23·4 2 2 35 2 · 8·27·7·19 b9·2 36 2 3 · 2·3 37 22 3 · 2·22·3 38 2·4 3·5 · 2·4·3·5 5 5 39 2 3 16·3·25·11 b10·2 ·3 40 24·4 3 2 24·4·3 41 23 · 22 2·24 42 2·8 3 2 22·8·3

43 · · 8·3·23 b11 44 8 ·· 8 45 23·16 9·5 · 2·23·16·9·5 46 24 3 · 24·3 3 3 47 2 ·4 3 32·9·5·7·13 b12·2 ·4·3 48 23·4 · 2 23·4 49 · 3 22 2·2·3 50 22 3 2 23·3

51 2·8 · 8·3 b13·2·8 52 23 3 · 23·3 53 24 · · 2·24 54 2·4 · · 2·4

55 · 3 16·3·5·29 b14·3 56 ·· 2 · 57 2 · 22 2·22 58 2 · 2 22 2 2 59 2 · 8·9·7·11·31 b15·2 60 4 · · 4 61 · · · ·

j j n stands for Z/n, n·m stands for Z/n ⊕ Z/m, and n stands for (Z/n) . Underlined symbols indicate contribu- bp 2k−2 2k−1 tions from Θn , and bk stands for 2 (2 − 1) times the numerator of 8ζ(1 − 2k), where ζ is the Riemann zeta function.

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MATHEMATICS extending from dimension 61 to dimension 90, the overall size of the computation more than doubles. Conjecture 1.2. Let f (k) be the product of the orders of the 2- primary stable homotopy groups in dimensions 1 through k. Then log f (k) = O(k 2). One interpretation of this conjecture is that the expected value of the logarithm of the order of the 2-primary component of πk grows linearly in k. 2. Groups of Homotopy Spheres An important application of stable homotopy group computations is to the work of Kervaire and Milnor (24) on the classification of smooth structures on spheres in dimensions at least 5. Let Θn be the group of h-cobordism classes of homotopy n-spheres. This group classifies the differential structures n bp on S for n ≥ 5. It has a subgroup Θn , which consists of homotopy spheres that bound parallelizable manifolds. See also ref. 25 for a survey on this subject. bp Kervaire and Milnor (24) showed that Θn is cyclic, and they determined the order of this group in terms of Bernoulli numbers. They also showed how Θn can be determined via bp exact sequences from Θn , the cokernel πn /J of the J - homomorphism, and the Kervaire invariant. See 3. v1-Periodic Stable Homotopy Groups for further discussion of the cokernel of J . Despite the recent breakthrough of Hill, Hopkins, and Ravenel (26), there is still one remaining unknown value of the Kervaire invariant in dimension 126. See the introduction of ref. 27 for a more detailed discussion of these ideas. Theorem 2.1. The last columns of Tables 1 and 2 describe the groups Θn for n ≤ 90, with the exception of n = 4. The bp underlined symbols denote the contributions from Θn , and bk is 22k−2(22k−1 − 1) times the numerator of 8ζ(1 − 2k), where ζ is the Riemann zeta function. The first few values, and then estimates, of the numbers bk (for k ≥ 2) are given by the sequence

28, 992, 8128, 261632, 1.45 × 109, 6.71 × 107, 1.94 × 1012, 7.54 × 1014, ....

We restate the following conjecture from ref. 27, which is based on the current knowledge of stable homotopy groups and a problem proposed by Milnor (25). Conjecture 2.2. In dimensions greater than 4, the only spheres with unique smooth structures are S 5, S 6, S 12, S 56, and S 61. Fig. 1. The 2-primary stable homotopy groups. Uniqueness in dimensions 5, 6, and 12 was known to Kervaire and Milnor (24). Uniqueness in dimension 56 is established in ref. 28 and uniqueness in dimension 61 is established in ref. 27. Fig. 1 displays the 2-primary stable homotopy groups in a Conjecture 2.2 is equivalent to the claim that the group Θn is graphical format. Vertical chains of n dots in column k indi- not of order 1 for dimensions greater than 61. This conjecture has n ∧ cate a copy of Z/2 in πk . The nonvertical lines indicate been confirmed in all odd dimensions by the second and the third additional multiplicative structure that we will not discuss here. authors (27) based on the work of Hill, Hopkins, and Ravenel The blue dots represent the v1-periodic subgroups. The green (26) and in even dimensions up to 140 by Behrens, Hill, Hopkins, dots are associated to the topological modular forms spectrum and Mahowald (29). tmf; the precise relationship is too complicated to describe here. Finally, the red dots indicate uncertainties. This type of graphical 3. v1-Periodic Stable Homotopy Groups description of stable homotopy groups is a modification by Allen Adams provided the first infinite families of elements in the sta- Hatcher of Adams spectral sequence charts (23). ble homotopy groups (30). Within the stable homotopy groups, The orders of individual 2-primary stable homotopy groups there is a regular repeating pattern of subgroups. These sub- do not follow a clear pattern, with large increases and decreases groups are known as the “v1-periodic stable homotopy groups,” seemingly at random. However, an empirically observed pattern and they are closely related to the “image of J .” emerges if we consider the cumulative size of the groups, i.e., the Theorem 3.1. Table 3 gives the v1-periodic stable homotopy ∧ product of the orders of all 2-primary stable homotopy groups in groups inside of πk for all k ≥ 2. Here ν(k + 1) is the 2-primary dimensions 1 through k. factor of the integer k + 1 (30). Our data strongly suggest that asymptotically, there is a lin- The odd-primary v1-periodic stable homotopy groups can be ear relationship between k 2 and the logarithm of this product of similarly described using odd-primary factors of integers. orders. In other words, the number of dots in Fig. 1 in columns 1 The v1-periodic subgroups are direct summands of the stable through k is asymptotically linearly proportional to k 2. Thus, in homotopy groups. Their complementary summands are known

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MATHEMATICS This recovers the classical dual Steenrod algebra, where τi and theory (51), after completing at a prime p. From this perspec- 2 ξi+1 correspond to ζi and ζi , respectively. tive, the generic fiber of C-motivic stable homotopy theory is On the other hand, after setting τ equal to zero, the result classical stable homotopy theory, and the special fiber has an is an exterior algebra on generators τi tensored with a polyno- entirely algebraic description. The special fiber is the category mial algebra on generators ξi+1. This structure is analogous to of BP∗BP-comodules, or equivalently, the category of quasico- Milnor’s description of the classical dual Steenrod algebra at odd herent sheaves on the moduli stack of one-dimensional formal primes (36). groups.

As in the classical case, ExtAC (F2[τ], F2[τ]) can be computed Theorem 6.1 is particularly valuable for computation. It means by machine in a large range. Then C-motivic Adams differentials that the Adams spectral sequence for S/τ can be computed in can be determined by the standard methods. As in the classi- an entirely algebraic manner, i.e., can be computed by machine cal case, dimension 61 seems to be the practical limit of this in a large range. This observation leads to the following innova- approach. tive program for computing classical stable homotopy groups: 1) Compute the C-motivic Adams E2-page by machine in a large 6. Algebraicity of the Cofiber of τ range. 2) Compute the algebraic Novikov spectral sequence by 0,−1 0,0 machine in a large range, including all differentials and multi- There is a map S → S in the C-motivic stable homotopy plicative structure. 3) Use Theorem 6.1 to deduce the structure of category that induces multiplication by τ in C-motivic cohomology. Therefore, we use the same notation τ for this map between the motivic Adams spectral sequence for S/τ. 4) Use the cofiber spheres. Let S/τ be the mapping cone (or cofiber) of τ. Surpris- sequence 0,−1 τ 0,0 1,−1 ingly, this stable two-cell complex turns out to have a remarkably S → S → S/τ → S algebraic structure. and naturality of Adams spectral sequences to pull back and push The Adams–Novikov spectral sequence is a remarkably effec- forward Adams differentials for S/τ to Adams differentials for tive tool for computing stable homotopy groups (5). The the motivic sphere. 5) Apply a variety of ad hoc arguments to input for this spectral sequence is the purely algebraic object deduce additional Adams differentials for the motivic sphere. 6) ExtBP∗BP (BP∗, BP∗), in which BP is the Brown–Peterson spec- Invert τ to obtain the classical Adams spectral sequence and the trum (49) and BP∗BP is the dual of the ring of its stable classical stable homotopy groups. operations. The algebraic input itself is complicated and can be The machine-generated data that we use in steps 1 and 2 are computed in a range by machine or by hand using the algebraic available at ref. 41. Novikov spectral sequence (18, 50). As the dimension increases, the ad hoc arguments of step 5 Theorem 6.1. The C-motivic Adams spectral sequence that com- become more and more complicated. Eventually, this approach putes the motivic stable homotopy groups of S/τ is isomorphic to will break down when the ad hoc arguments become too the algebraic Novikov spectral sequence (51). complicated to resolve. It is not yet clear when that will occur. In fact, Theorem 6.1 is a computational corollary of other more structural results. In particular, Gheorghe demonstrated Data Availability. Spreadsheets, source code, and documentation have that the C-motivic spectrum S/τ is an E∞-ring object in an been deposited in GitHub (https://github.com/pouiyter/morestablestems essentially unique way (52), and the homotopy category of and https://github.com/pouiyter/MinimalResolution). cellular S/τ-modules is equivalent to a derived category of BP∗BP-comodules (51). ACKNOWLEDGMENTS. D.C.I. was supported by NSF grant DMS-1606290. G.W. was supported by grant NSFC-11801082 and by the Shanghai Rising- Deformation theory provides a unifying perspective on this Star Program. Z.X. was supported by NSF grants DMS-1810638 and DMS- circle of ideas. The key insight is that C-motivic cellular stable 2043485. Many of the associated machine computations were performed homotopy theory is a deformation of classical stable homotopy on the Wayne State University Grid high-performance computing cluster.

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